Question 6: Verify the properties for the sets \(A\), \(B\) and \(C\) given below:
\((b)\) \(\;\) \(A=\phi\), \(B=\{0\}\), \(C=\{0,1,2\}\)
\((i)\) Associative of Union
\((ii)\) Associativity of intersection
\((iii)\) Distributive of Union over intersection
\((iv)\) Distributive of intersection over union
Solution:
\((i)\) Associative of Union
\((A\cup B)\cup C=A\cup (B\cup C)\) is Associative of Union
\(\text{L.H.S.}=(A\cup B)\cup C\)
\(\ \ \ \ \ \ \ \ \ \ \ =[\{\}\)\(\cup\)\(\{0\}]\)\(\cup\)\(\{0,1,2\}\)
\(\ \ \ \ \ \ \ \ \ \ \ =\{0\}\)\(\cup\)\(\{0,1,2\}\)
\(\ \ \ \ \ \ \ \ \ \ \ =\{0,1,2\}\)
\(\text{R.H.S.}=A\cup (B\cup C)\)
\(\ \ \ \ \ \ \ \ \ \ \ =\{\}\)\(\cup\)\([\{0\}\)\(\cup\)\(\{0,1,2\}]\)
\(\ \ \ \ \ \ \ \ \ \ \ =\{\}\)\(\cup\)\(\{0,1,2\}\)
\(\ \ \ \ \ \ \ \ \ \ \ =\{0,1,2\}\)
Hence, \(\text{L.H.S.}=\text{R.H.S.}\)
\((ii)\) Associativity of intersection
\((A\cap B)\cap C=A\cap (B\cap C)\) is Associative of Intersection
\(\text{L.H.S.}=(A\cap B)\cap C\)
\(\ \ \ \ \ \ \ \ \ \ \ =[\{\}\)\(\cap\)\(\{0\}]\)\(\cap\)\(\{0,1,2\}\)
\(\ \ \ \ \ \ \ \ \ \ \ =\{\}\)\(\cap\)\(\{0,1,2\}\)
\(\ \ \ \ \ \ \ \ \ \ \ =\{\}\)
\(\text{R.H.S.}=A\cap (B\cap C)\)
\(\ \ \ \ \ \ \ \ \ \ \ =\{\}\)\(\cap\)\([\{0\}\)\(\cap\)\(\{0,1,2\}]\)
\(\ \ \ \ \ \ \ \ \ \ \ =\{\}\)\(\cap\)\(\{0\}\)
\(\ \ \ \ \ \ \ \ \ \ \ =\{\}\)
Hence, \(\text{L.H.S.}=\text{R.H.S.}\)
\((iii)\) Distributive of Union over intersection
\(A \cup (B \cap C) = (A \cup B) \cap (A \cup C)\) is Distributive of Union over intersection
\(\text{L.H.S.}=A \cup (B \cap C)\)
\(\ \ \ \ \ \ \ \ \ \ \ =\{\}\)\(\cup\)\([\{0\}\)\(\cap\)\(\{0,1,2\}]\)
\(\ \ \ \ \ \ \ \ \ \ \ =\{\}\)\(\cup\)\(\{0\}\)
\(\ \ \ \ \ \ \ \ \ \ \ =\{0\}\)
\(\text{R.H.S.}=(A \cup B) \cap (A \cup C)\)
\(\ \ \ \ \ \ \ \ \ \ \ =[\{\}\)\(\cup\)\(\{0\}]\)\(\cap\)\([\{\}\)\(\cup\)\(\{0,1,2\}]\)
\(\ \ \ \ \ \ \ \ \ \ \ =\{0\}\)\(\cap\)\(\{0,1,2\}\)
\(\ \ \ \ \ \ \ \ \ \ \ =\{0\}\)
Hence, \(\text{L.H.S.}=\text{R.H.S.}\)
\((iv)\) Distributive of intersection over union
\(A \cap (B \cup C) = (A \cap B) \cup (A \cap C)\) is Distributive of intersection over union
\(\text{L.H.S.}=A \cap (B \cup C)\)
\(\ \ \ \ \ \ \ \ \ \ \ =\{\}\)\(\cap\)\([\{0\}\)\(\cup\)\(\{0,1,2\}]\)
\(\ \ \ \ \ \ \ \ \ \ \ =\{\}\)\(\cap\)\(\{0,1,2\}\)
\(\ \ \ \ \ \ \ \ \ \ \ =\{\}\)
\(\text{R.H.S.}=(A \cap B) \cup (A \cap C)\)
\(\ \ \ \ \ \ \ \ \ \ \ =[\{\}\)\(\cap\)\(\{0\}]\)\(\cup\)\([\{\}\)\(\cap\)\(\{0,1,2\}]\)
\(\ \ \ \ \ \ \ \ \ \ \ =\{\}\)\(\cup\)\(\{\}\)
\(\ \ \ \ \ \ \ \ \ \ \ =\{\}\)
Hence, \(\text{L.H.S.}=\text{R.H.S.}\)